The Nature of Mathematical Modelling

6.1

(a) Work out the first three cumulants C1, C2, and C3.

(b) Evaluate the first three cumulants for a Gaussian distribution

6.2

(a) If ~y(~x) = (y1(x1, x2), y2(x1, x2)) is a coordinate transformation, what is the area of a differential element dx1 dx2 after it is mapped into the ~y plane? Recall that the area of a parallelogram is equal to the length of its base times its height.

6.4

(a) Use a Fourier transform to solve the diffusion equation (6.57) (assume that the initial condition is a normalized delta function at the origin).

(b) What is the variance as a function of time?

(c) How is the diffusion coefficient for Brownian motion related to the viscosity of a fluid?

(d) Write a program (including the random number generator) to plot the position as a function of time of a random walker in 1D that at each time step has an equal probability of making a step of ± 1. Plot an ensemble of 10 trajectories, each 1000 points long, and overlay error bars of width 3σ(t) on the plot.